The threshold interpolation conjecture for consecutive patterns in random permutations

Let $\boldsymbol{\sigma}_{n,m}$ be a uniformly random permutation of $[n]$ with exactly $m$ inversions. For a permutation $\rho$, write $\rho$ for its complement by replacing each entry $i$ with $|\rho|+1-i$, and let $\boldsymbol{\sigma}_{n,m}$ contain $\rho$ mean that $\rho$ occurs as a consecutive pattern. For a permutation $\rho$, let $\operatorname{inv}(\rho)$ denote its number of inversions. Let $\pi$ be any consecutive permutation pattern, and set

$$s=\operatorname{inv}(\pi),\qquad s'=\operatorname{inv}(\overline{\pi}).$$

The threshold interpolation conjecture. If $n^{1-1/s}\ll m$ and $\binom{n}{2}-m\gg n^{1-1/s'}$, then

$$\lim_{n\to\infty}\mathbb{P}\big[\boldsymbol{\sigma}_{n,m}\text{ contains }\pi\big]=1.$$

The preceding theorem establishes the appearance and disappearance thresholds separately, while this conjecture asserts almost-sure presence throughout the interval between them. The paper notes that its random-composition methods do not establish this intermediate regime.